It turns out that, when matter is not taken into account, any ST theory becomes equivalent to some \(f(R)\)  gravity.
Any (non-degenerate) STT in the Einstein frame has \(\bar{{\mathcal {B}}}={1\over 2}\)  or \(\bar{{\mathcal {B}}}=-{1\over 2}\) . The first case corresponds to metric \(f(R)\)  theory and to a sub-class of hybrid metric-Palatini gravity. Following our earlier paper [1]}, the
second case corresponds to a different sub-class of hybrid metric-Palatini formulation (discussed below).
Finally, in degenerate case \(\bar{{\mathcal {B}}}=0\)  not studied in this paper, one can find the relation with purely Palatini \(f(R)\) -theory. This corollary follows directly from analysis of the sign of \(\mathcal {I}\) :
\(\text{sign}(\mathcal {I}) = -\text{sign}(m)={\left\lbrace \begin{array}{ll}+1& \text{ metric and hybrid}, \\0 &\text{ Palatini}, \\-1& \text{ hybrid},\end{array}\right.}\) 
